Worst case model risk management

Denis Talay and Ziyu ZhengINRIA

2004 route des LuciolesBP93

F-06902 Sophia Antipolis cedex (France)

Abstract: We are interested in model risk control problems. We studya strategy for the trader which, in a sense, guarantees good performanceswhatever is the unknown model for the assets of his/her portfolio. Thetrader chooses trading strategies to decrease the risk and therefore acts as aminimizer; the market systematically acts against the interest of the trader,so that we consider it acts as a maximizer. Thus we consider the model riskcontrol problem as a two players (Trader versus Market) zero-sum stochasticdifferential game problem. Therefore our construction corresponds to a ‘worstcase’ worry and, in this sense, can be viewed as a continuous-time extensionof discrete-time strategies based upon prescriptions issued from VaR analysesat the beginning of each period. In addition, the initial value of the optimalportfolio can be seen as the minimal amount of money which is needed toface the worst possible damage.

We give a proper mathematical statement for such a game problem. Weprove that the value function of this game problem is the unique viscositysolution to an Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, and satisfiesthe Dynamic Programming Principle.

Key words: Model risk, stochastic differential game, Hamilton-Jacobi-Bellman-Isaacs equation.

JEL Classification: G11Mathematics Subject Classification (1991): 90A09, 60H30, 93E05.

1

1 Introduction

In this note we describe the financial strategy which a trader can follow inorder to manage his/her model risk.

Suppose that the trader precisely knows the model followed by the realmarket, and that this model is given by a system of stochastic differentialequations. Then, in a complete market, she/he is able to construct a strategywhich perfectly hedges the option. For options written on discount bonds,such an assumption means that the trader knows perfectly the descriptionof the random evolution of the term structure of the interest rates. Thisis unrealistic for various reasons, for example: lack of sufficient informationfrom the market, choice of the number of factors which drive the term struc-ture of interest rates, choice of the modelling stochastic processes, estimatesof the parameters required to use the model, complexity of the estimationprocedure, etc.

When one has a rather precise information on the model of the mar-ket, then one can take advantage of the robustness of formulae of Black andScholes type (see, e.g., El Karoui, Jeanblanc–Picque & Shreve [8] and Ro-magnoli & Vargiolu [15]). When one has only a vague information on themodel of the market, can one find a strategy, which, in a sense, guaranteestolerable performances whatever the unknown model is? To address thisquestion several authors have proposed super–replication techniques: see,e.g., Avellaneda, Levy & Paras [2], Avellaneda & Paras [3], Touzi [17] andreferences therein. Here we adopt a ‘worst case’ stochastic game approachwhich can be summarized as follows:

Trader = Minimizer of Risk.

Market = Maximizer of Risk.

Trader vs Market.

We introduce a cost function which describes the risk faced by the trader.Instead of supposing that the trader knows the exact model followed by thereal market, we assume that the trader knows that the correct model ofthe market belongs to a wide class of models. The trader chooses tradingstrategies from a set of admissible strategies to decrease the risk and thereforeacts as a minimizer of the risk; on the other hand, we suppose that the marketsystematically behaves against the interest of the trader, and therefore weconsider it acts as a maximizer of the risk. Thus the model risk controlproblem can be set up as a two players (Trader versus Market) zero-sumstochastic differential game problem and the corresponding strategies, in asense, are continuous-time versions of discrete-time procedures based uponprescriptions issued from VaR analyses at the beginning of each period.

2

In Section 2, we introduce our framework: the misspecified model forthe asset prices and our definition of model risk. In Section 3, we state thestochastic differential game problem which is the key tool of our approach;we slightly extend results by Fleming & Souganidis [10] to prove that thevalue function of this game problem is the unique viscosity solution to anHamilton-Jacobi-Bellman-Isaacs equation; we also prove that the value func-tion satisfies the Dynamic Programming Principle. In Section 4, we illustrateour methodology by considering the particular case of the hedging of Euro-pean bond options. We conclude by showing a numerical result.

Our paper is deeply related to the paper [7] by Cvitanic and Karatzaswho have introduced and studied dynamic measure of risks such as

infπ(·)∈A(x)

supν∈D

E ν(F (Xx,π(T ))),

where A(x) denotes the class of admissible portfolio strategies, and E ν de-notes the expectation under the probability P ν for all ν in a suitable set. Allthe measures P ν have the same risk-neutral equivalent martingale measure,which implies that the trader (or the regulator) is concerned by model riskon stock appreciation rates. Here, we include model risk on volatilities andprove the relationship between the dynamic hedging under the worst casescenario and viscosity solutions of fully nonlinear PDEs, which allows us todevelop numerical methods. We thus hopefully answer to a part of the openproblems listed in the Conclusion in [7].

We finally emphasize that the solution at time 0 of our stochastic gameproblem can be viewed as the minimal amount of money that the financialinstitution needs to “contain the worst possible damage” as said in [7].

2 Our financial model

Suppose that the financial market consists in n risky financial assets, whoseprices at time t are denoted by Pi(t), and one instantaneously risk-free asset,the money market, whose price at time t is denoted by P0(t). We use thefollowing stochastic differential equation to describe this financial market.

dPi(t) = Pi(t)[bi(t)dt+∑d

j=1 σij(t)dWjt ], 1 ≤ i ≤ n,

dP0(t) = r(t)P0(t)dt.(1)

Here we consider a probability space (Ω,F ,P ), and processes bi(·), σij(·),r(·) which are progressively measurable with respect to the P augmentedfiltration of the d dimensional Brownian motion (Wt). We also suppose thatthere exists a unique strong solution to (1).

3

Consider an economic agent whose decisions do not affect the prices inthe market (a small investor). We denote by X(t) the wealth of this agentat time t, by Hi(t) the amount he/she invests in the ith risky financial assetat that time (1 ≤ i ≤ n), and by H0(t) the amount he/she invests in therisk-free asset. Then the value of his/her portfolio is

X(t) =n∑

i=1

Hi(t)Pi(t) +H0(t)P0(t).

Suppose that the portfolio is self-financing. Then the process X(t) satisfiesthe following stochastic differential equation

dX(t) =n∑

i=1

Hi(t)Pi(t)

[bi(t)dt+

d∑j=1

σij(t)dWt

]+H0(t)r(t)P0(t)dt.

If X(t) 6= 0, set

πi(t) :=Hi(t)Pi(t)

X(t), i = 1, . . . , n,

and

π0(t) :=H0(t)P0(t)

X(t).

If X(t) = 0, set πi(t) = π0(t) = 0. With the above notation, we have

dX(t) = X(t)n∑

i=1

πi(t)

[bi(t)dt+

d∑j=1

σij(t)dWjt

]+X(t)r(t)

(1−

n∑i=1

πi(t)

)dt.

(2)

3 Model risk control

To simplify the notation, in this section we suppose that the process (r(t))is constant and equal to r > 0. It is straightforward to include it in the listof the controls, or to more specifically include model risks on a fixed type ofyield curves models (see Section 4 for such an example).

Suppose that the market chooses the risk premium and the volatilityprocess to increase the risk of the position.

We adopt the definition of admissible controls and strategies of Fleming& Souganidis [10] and, as in [10], we introduce the canonical sample spacesfor the underlying Brownian motion in (1) and (2). For each θ ∈ [0, T ] weset

Ωθ := (ω ∈ C([θ, T ]; Rd) : ωθ = 0).

4

We denote by (Fθ,s) the filtration generated by the canonical process fromtime θ to time s. Equipped with the Wiener measure P θ on Fθ,T , the filteredprobability space (Ωθ,Fθ,T ,P θ, (Fθ,s, θ ≤ s ≤ T )) is the canonical samplespace.

We define the admissible controls and the admissible strategies by:

Definition 3.1. An admissible control process u(·) := (b(·), σ(·)) for themarket on [θ, T ] is an (Fθ,s)-progressively measurable process taking valuein a compact subset Ku of Rn × Rnd. An admissible control process π(·)for the investor on [θ, T ] is an (Fθ,s)-progressively measurable process tak-ing value in a compact subset Kπ of Rn. The set of all admissible controlsfor the market on [θ, T ] is denoted by Adu(θ) and the set of all admissi-ble controls for the investor on [θ, T ] is denoted by Adπ(θ). We say thattwo admissible control processes u and u in Adu(θ) are the same on [θ, s]if Pθ(u(·) = u(·) a.e. in [θ, s]) = 1. A similar convention is assumed forelements of Adπ(θ). We write u(·) ≈ u(·) on [θ, s] (respectively, π(·) ≈ π(·)on [θ, s]) when u and u (respectively, π and π) are the same on [θ, s].

Definition 3.2. An admissible strategy Π for the investor on [θ, T ] is amapping Π : Adu(θ) → Adπ(θ) such that if u(·) ≈ u(·) on [θ, s], thenΠ(u(·))(·) ≈ Π(u(·))(·) on [θ, s] for every s ∈ [θ, T ]. The set of all admissiblestrategies for the investor on [θ, T ] is denoted by AdΠ(θ).

For given Π ∈ AdΠ(θ) and u(·) = (b(·), σ(·)) ∈ Adu(θ), we set π(·) :=Π(u(·))(·).

The controlled system of prices and value of the portfolio is defined asfollows:dPi(t) = Pi(t)[bi(t)dt+

∑dj=1 σij(t)dW

jt ] for 0 ≤ i ≤ n,

dX(t) = X(t)∑n

i=1 πi(t)[bi(t)dt+

∑dj=1 σij(t)dW

jt

]+ rX(t) (1−

∑ni=1 πi(t)) dt.

(3)Given a suitable function F the cost function is now defined as follows:

J(θ, p, x,Π, u(·)) := E θ,p,xF (P (T ), X(T )), (4)

where the symbol E θ,p,x stands for the conditional expectation under P θ

knowing that (P (θ) = p,X(θ) = x)1. In this section we do not discuss thechoice of F in practice, which highly depends on the financial objective of thetrader or of the regulator. We thus make a weak assumption on F (see (8)below), and postpone some discussion to Section 4.

We then introduce the value function of our game problem.

1For the sake of simplicity our notation does not emphasize that the process (P (t), X(t))is parametered by (b(t), σ(t), πt).

5

Definition 3.3. The lower value of the model risk control problem with initialdata (θ, p, x) is

V (θ, p, x) := infΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

J(θ, p, x,Π, u(·)). (5)

3.1 Our main result

Define the (n+ 1)× (n+ 1) symmetric matrix a(p, x, σ, π) asaij(p, x, σ, π) :=

∑dk=1(piσikpjσjk) for 1 ≤ i, j ≤ n,

an+1,j(p, x, σ, π) :=∑d

k=1

∑nl=1(xπlσklpjσjk) for 1 ≤ j ≤ n,

an+1,n+1(p, x, σ, π) :=∑d

k=1

∑nl=1(x

2π2l σ

2kl).

(6)

Let q(p, x, b, π) denote the n+ 1 dimensional vector

q(p, x, b, π) :=

(p1b1, . . . , pnbn, x

(r +

d∑i=1

πi(bi − r)

)). (7)

For all u = (b, σ) ∈ Ku and π ∈ Kπ we set

Hu,π(A, z, p, x) :=

[1

2Tr (a(p, x, σ, π)A) + z · q(p, x, b, π)

],

andH−(A, z, p, x) := max

u∈Ku

minπ∈Kπ

[Hu,π(A, z, p, x)] .

Our main result is

Theorem 3.4. Suppose that F is a continuous function such that

|F (p, x)− F (p, x)| ≤ Q(|p|, |x|, |p|, |x|)(|p− p|+ |x− x|), (8)

where Q(|p|, |x|, |p|, |x|) is a polynomial function.Then the value function V (θ, p, x) defined in (5) is the unique viscosity

solution in the space

S := ϕ(t, p, x) is continuous on [0, T ]× Rn × R; ∃A > 0,

lim|p|2+x2→∞

ϕ(t, p, x) exp(−A| log(|p|2 + x2)|2) = 0 for all t ∈ [0, T ]

to the Hamilton-Jacobi-Bellman-Isaacs Equation∂v

∂t(t, p, x) +H−(D2v(t, p, x), Dv(t, p, x), p, x) = 0 in [0, T )× Rn+1,

v(T, p, x) = F (p, x).

(9)

6

In addition, V (θ, p, x) satisfies the Dynamic Programming Principle, that is,

V (θ, p, x) = infΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x[V (t, P (t), X(t))] for all θ < t < T.

(10)

3.2 Proof of Theorem 3.4

We start the proof of Theorem 3.4 by an easy lemma which implies that thevalue function V belongs to the space S.

Lemma 3.5. V (θ, p, x) is time continuous and locally Lipschitz continuousw.r.t. (p, x) in Rn+1. There exists a positive polynomial function Q of (p, x)such that |V (θ, p, x)| ≤ Q(p, x) for all θ in [0, T ].

Proof. The continuity in θ of V results from the classical estimates on thesolution of stochastic differential equations: see, e.g., Fleming & Soner [9,Chap.IV, Sec.6].

We now check that, for each pair of admissible control and strategy,the cost function is locally Lipschitz continuous w.r.t. (p, x) in Rn+1. Let(P (t), X(t)) (respectively (P (t), X(t))) be the solution to (3) with initialcondition (p, x) (respectively (p, x)) at time θ. In view of (8), we have

|F (P (T ), X(T ))− F (P (T ), X(T ))|≤ Q(|P (T )|, |X(T )|, |P (T )|, |X(T )|)(|P (T )− P (T )|+ |X(T )−X(T )|).

An elementary calculation and classical estimates lead to∣∣E θ,p,x[F (P (T ), X(T ))]− E θ,p,x[F (P (T ), X(T ))]∣∣

≤ Q(|p|, |x|, |p|, |x|)√

(|p− p|2 + |x− x|2),

for some polynomial function Q. The Lipschitz continuity of V w.r.t. (p, x)is a then straightforward consequence of the following inequalities: for allbounded functions f1, f2 and set Z,

| supz∈Z f1(z)− supz∈Z f2(z)| ≤ supz∈Z |f1(z)− f2(z)|,| infz∈Z f1(z)− infz∈Z f2(z)| ≤ supz∈Z |f1(z)− f2(z)|.

(11)

A similar computation proves the polynomial growth in (p, x) of V (θ, p, x).

If the controlled system had bounded coefficients and F were a boundedLipschitz function, Theorem 2.4 would result from Fleming & Souganidis [10,Thm.1.6,Thm.2.6]. As the coefficients of (3) are unbounded and F is locallyLipschitz only, we have to use localization techniques.

7

3.2.1 Existence of a viscosity solution

We set Bk := (p, x) ∈ Rn+1, |p|2 + x2 < k2. Choose a function φk inC∞b (Rn+1) such that φk(p, x) = 1 on Bk, and φk(p, x) = 0 outside Bk+1, andthe Lipschitz constant of φk is less than 2. Consider the following S.D.EdP k

i (t) = φk(Pk(t), Xk(t))P k

i (t)[bi(t)dt+∑d

j=1 σij(t)dWjt ] for 0 ≤ i ≤ n,

dXk(t) = φk(Pk(t), Xk(t))

[Xk(t)

∑ni=1 πi(t)

[bi(t)dt+

∑dj=1 σij(t)dW

jt

]+rXk(t)(1−

∑ni=1 πi(t))dt

].

(12)Observe that the coefficients of (12) are uniformly Lipschitz and bounded.

Our objective is to use the stability lemma for viscosity solutions (Li-ons [12, Prop.I.3]) in order to construct a viscosity solution of Equation (9).Define a cost function Jk as in (4) and a lower value function V k(θ, p, x) asin (5) by substituting (P k(t), Xk(t)) to (P (t), X(t)) andF k(p, x) := φk+2(p, x)F (p, x) to F (p, x).

Consider the HJBI equation associated with (12), that is,∂v

∂t(t, p, x) +H−

k (D2v(t, p, x), Dv(t, p, x), p, x) = 0 in [0, T )× Rn+1,

v(T, p, x) = F k(p, x),

(13)where, the functions a, q being defined in (6) and (7),

H−k (A, z, p, x)

:= maxu∈Ku

minπ∈Kπ

[1

2Tr(φk(p, x)

2a(p, x, σ, π)A)

+ φk(p, x)z · q(p, x, b, π)

]. (14)

All the assumptions of Fleming & Souganidis [10, Thm.2.6] are satisfied, sothat V k is the unique viscosity solution to (13). In addition, as the righthand side of (12) is null as soon as (P k(t), Xk(t)) is outside Bk+1, and asF k(p, x) ≡ F (p, x) inside Bk+2, in view of the definition of V k as a valuefunction one has

V k(θ, p, x) = F (p, x) (15)

for all θ ∈ [0, T ] and (p, x) ∈ Bk+2 −Bk+1.Set

V k(t, p, x) :=

V k(t, p, x) in [0, T ]×Bk+1,

F (p, x) in [0, T ]× Rn+1 −Bk+1.(16)

In view of (15) and the continuity of V k, V k is a continuous function on[0, T ] × Rn+1. Moreover, as F k ≡ F in Bk+1 and H−

k (A, z, p, x) = 0 for all

8

(p, x) in Rn+1 −Bk+1, we have that V k is a viscosity solution to∂v

∂t(t, p, x) +H−

k (D2v(t, p, x), Dv(t, p, x), p, x) = 0 in [0, T )× Rn+1,

v(T, p, x) = F (p, x).

(17)Moreover, in view of (12), for all (p, x) outside Bk+1 one has

E θ,p,xF (P k(T ), Xk(T )) = F (p, x).

Thus, for all (p, x) ∈ Rn+1 one has

V k(t, p, x) = infΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x[F (P k(T ), Xk(T ))].

Observe that H−k (A, z, p, x) converges to H−(A, z, p, x) locally on compact

subsets of S2 × Rn+1 × Rn+1. Therefore, owing to Lions [12, Prop.I.3], ifthe viscosity solutions V k to (17) converge on compact subsets of R+×Rn+1

to the function V , then V is a viscosity solution to (9). We now prove theconvergence.

Arbitrarily fix a compact subset of Rn+1. This set is included in the ballBk for all k large enough. Let (p, x) be in the compact set under consid-eration. Fix an arbitrary admissible control u· and an arbitrary admissiblestrategy Π. Let τk be the minimum of T and the first exit time of (P (t), X(t))from Bk. Before the stopping time τk, the solutions to (3) and (12) coincide.Thus

|E θ,p,x[F (P (T ), X(T ))− F (P k(T ), Xk(T ))]|= |E θ,p,x[F (P (T ), X(T ))− F (P k(T ), Xk(T ))]Iτk<T ]|≤ E θ,p,x

[Q(|P (T )|, |X(T )|, |P k(T )|, |Xk(T )|)

(|P (T )|+ |X(T )|+ |P k(T )|+ |Xk(T )|)Iτk<T

].

From

E θ,p,x[|P k(T )|2m + (Xk(T ))2m + |P (T )|2m + (X(T ))2m] ≤ C(T )(|p|2m + x2m),(18)

and

P θ,p,x(τk < T ) = P θ,p,x

[sup

θ≤t≤T(|P (t)|2 +X(t)2) ≥ k2

]≤ (|p|2 + x2)C(T )

k2,

(19)

9

for some constant C(T ) independent from k, using Cauchy–Schwarz’s in-equality we deduce

|E θ,p,x[F (P (T ), X(T ))− F (P k(T ), Xk(T ))]| ≤ Φ(|p|, |x|)k

,

where Φ is a polynomial function determined by the set Ku×Kπ, the functionF , and the time T only. Using Inequality (11), for all θ in [0, T ] and (p, x)in the compact set under consideration, we deduce that

|V k(θ, p, x)− V (θ, p, x)| ≤ Φ(|p|, |x|)k

. (20)

Thus V k converges to V uniformly on compact subsets of R+ × Rn+1, fromwhich we deduce that V is a viscosity solution to (9).

3.2.2 Uniqueness of a viscosity solution

The next statement and its proof are appropriate modifications to the unique-ness result of Barles, Buckdahn & Pardoux [4, Thm.3.5].

Theorem 3.6. Suppose that there exist a viscosity subsolution v(t, p, x) anda viscosity supersolution w(t, p, x) to (9) such that

lim|p|2+x2→∞

v(t, p, x) exp(−A| log(|p|2 + x2)|2) = 0 for all t ∈ [0, T ],

and

lim|p|2+x2→∞

w(t, p, x) exp(−A| log(|p|2 + x2)|2) = 0 for all t ∈ [0, T ]

for some A > 0.Moreover suppose that v(T, p, x) ≤ w(T, p, x) for all (p, x) ∈ Rn+1. Then

v(t, p, x) ≤ w(t, p, x) for all (t, p, x) ∈ [0, T ]× Rn+1.

We start with two technical lemmas.

Lemma 3.7. Let v(t, p, x) be a viscosity subsolution and w(t, p, x) a viscositysupersolution to (9). Then the function v := v −w is a viscosity subsolutionto

∂v

∂t(t, p, x) +H+(D2v(t, p, x), Dv(t, p, x), p, x) = 0 in [0, T )× Rn+1, (21)

where

H+(A, z, p, x) := maxu=(b,σ)∈Ku

maxπ∈Kπ

(1

2Tr (a(p, x, σ, π)A) + z · q(p, x, b, π)

),

for all (n+ 1)× (n+ 1) symmetric matrix A and all vector z in Rn+1.

10

Proof. Let ϕ ∈ C2([0, T ]×Rn+1) and let (t, p, x) be a strict global maximumpoint of v − ϕ.

We introduce the following function

ψε,α(t, p, x, s, p, x)

:= v(t, p, x)− w(s, p, x)− |p− p|2 + |x− x|2

ε2− |t− s|2

α2− ϕ(t, p, x),

where ε, α are positive real numbers which are devoted to tend to zero.By a standard argument in viscosity solution theory, (see Crandall, Ishii

& Lions [6]), there exists a sequence (tε,α, pε,α, xε,α, sε,α, pε,α, xε,α) such that

• (tε,α, pε,α, xε,α, sε,α, pε,α, xε,α) is a global maximum point of ψε,α in ([0, T ]×BR)2 where BR is a large ball in Rn+1.

• (tε,α, pε,α, xε,α, sε,α, pε,α, xε,α) tends to (t, p, x, t, p, x) when (ε, α) → 0.

•|pε,α − pε,α|2 + |xε,α − xε,α|2

ε2+|tε,α − sε,α|2

α2is bounded and tends to zero

when (ε, α) → 0.

We now use the jet sets D2,+v(t, p, x) and D

2,−w(t, p, x). It follows from

Crandall, Ishii & Lions [6, Thm.8.3] that there exist X, Y ∈ S2 such that(aε,α +

∂ϕ

∂t(tε,α, pε,α, xε,α), zε,α +Dϕ(tε,α, pε,α, xε,α), X

)∈ D2,+

v(tε,α, pε,α, xε,α),

(aε,α, zε,α, Y ) ∈ D2,−w(sε,α, pε,α, xε,α),

and [X 00 −Y

]≤ 4

ε2

[I −I−I I

]+

[D2ϕ(tε,α, pε,α, xε,α) 0

0 0

],

where

aε,α :=2(tε,α − sε,α)

α2,

and

zε,α :=2((pε,α, xε,α)− (pε,α, xε,α))

ε2.

Observing that (tε,α, pε,α, xε,α, sε,α, pε,α, xε,α) is a local maximum point of ψε,α

and v(t, p, x) is a viscosity subsolution of (9), we obtain

aε,α +∂ϕ

∂t(tε,α, pε,α, xε,α) +H−(X, zε,α +Dϕ(tε,α, pε,α, xε,α), pε,α, xε,α) ≥ 0.

(22)

11

Similarly, since w(t, p, x) is a viscosity supersolution to (9), we have

aε,α +H−(Y, zε,α, pε,α, xε,α) ≤ 0. (23)

Thus

∂ϕ

∂t(tε,α, pε,α, xε,α) +H−(X, zε,α +Dϕ(tε,α, pε,α, xε,α), pε,α, xε,α)

−H−(Y, zε,α, pε,α, xε,α) ≥ 0. (24)

Suppose that we have shown

H−(X, zε,α +Dϕ(tε,α, pε,α, xε,α), pε,α, xε,α)−H−(Y, zε,α, pε,α, xε,α) (25)

≤ maxu∈Ku

maxπ∈Kπ

[Tr(a(pε,α, xε,α, σ, π)D2ϕ(tε,α, pε,α, xε,α))

+Dϕ(tε,α, pε,α, xε,α) · q(pε,α, xε,α, b, π)]

+ C|pε,α − pε,α|2 + |xε,α − xε,α|2

ε2

+ C| < zε,α, ((pε,α, xε,α)− (pε,α, xε,α)) > |

for some positive real number C. We use the right side of (25) in (24). Inview of the definition of (tε,α, pε,α, xε,α, pε,α, xε,α), the right side of (25) tendsto

maxu∈Ku

maxπ∈Kπ

[Tr(a(p, x, σ, π)D2ϕ(t, p, x)) +Dϕ(t, p, x) · q(p, x, b, π)

]when (ε, α) tends to zero. We then get our desired result:

∂ϕ

∂t(t, p, x) +H+(D2ϕ(t, p, x), Dϕ(t, p, x), p, x) ≥ 0 in [0, T )× Rn+1. (26)

We now prove (25). We have

H−(X, z, p, x)−H−(Y, z, p, x)

= maxu∈Ku

minπ∈Kπ

[Hu,π(X, z, p, x)]− maxu∈Ku

minπ∈Kπ

[Hu,π(Y, z, p, x)]

≤ maxu∈Ku

maxπ∈Kπ

(Hu,π(X, z, p, x)−Hu,π(Y, z, p, x))

≤ maxu∈Ku

maxπ∈Kπ

(Hu,π(X, z, p, x)−Hu,π(Y, z, p, x)

+ (Hu,π(X, z, p, x)−Hu,π(X, z, p, x))),

andHu,π(X, z, p, x)−Hu,π(X, z, p, x) =< q(p, x, b, π), z − z > .

12

In addition, from the definition of X,Y it results that[X −D2ϕ(tε,α, pε,α, xε,α) 0

0 −Y

]≤ 4

ε2

[I −I−I I

].

In view of Fleming & Soner [9, Chap.V, Lemma 6.2], there exists C > 0 suchthat

Hu,π(X, z, p, x)−Hu,π(Y, z, p, x)

≤ C|p− p|2 + |x− x|2

ε2+ Tr(a(p, x, σ, π)D2ϕ(tε,α, pε,α, xε,α))

+ < q(p, x, b, π)− q(p, x, b, π), z >

for all X, Y , z, (p, p) and (x, x). That ends the proof.

Lemma 3.8. Set

φ(p, x) :=

[1

2log(|p|2 + x2 + 1) + 1

]2

.

For any A > 0, there exists a C1 >AT

such that the function

ρ(t, p, x) := exp[(C1(T − t) + A)φ(p, x)],

satisfies

∂ρ

∂t(t, p, x) +H+(D2ρ(t, p, x), Dρ(t, p, x), p, x) < 0 in [t1, T )× Rn+1,

where t1 := T − A

C1

.

Proof. In view of the definition of t1, there exists a constant C independentof C1 such that, for all t in [t1, T ],

|Dρ(t, p, x)| ≤ Cρ(t, p, x)φ

12 (p, x)

(|p|2 + x2 + 1)12

,

|D2ρ(t, p, x)| ≤ Cρ(t, p, x)φ(p, x)

(|p|2 + x2 + 1).

As, in addition,∂ρ(t, p, x)

∂t= −C1ρ(t, p, x)φ(p, x),

13

we have

|H+(D2ρ(t, p, x), Dρ(t, p, x), p, x)|≤ C((|p|2 + x2)|D2ρ(t, p, x)|+ (|p|+ |x|)|Dρ(t, p, x)|)≤ Cρ(t, p, x)φ(p, x),

and the result follows by choosing a large enough C1.

In order to conclude the proof of Theorem 3.6, we now proceed in threesteps: we will first prove that

v(t, p, x) ≤ αρ(t, p, x) in [t1, T )× Rn+1

for any α > 0, where ρ and t1 are defined in Lemma 3.8. Then we will let αtend to zero, and finally we will show that

v(t, p, x) ≤ αρ(t, p, x) in [0, T )× Rn+1.

Since by assumption vαρ

tends to zero when |p|2 + x2 tends to infinity, we

have that v − αρ is bounded from above in [t1, T ] × Rn+1, and the globalmaximum

M := max[t1,T ]×Rn+1

(v − αρ)(t, p, x)

is achieved at some point (t0, p0, x0).We assume that v(t0, p0, x0) > 0, otherwise we are done.Because (t0, p0, x0) is the global maximum, we have

(v − αρ)(t, p, x) ≤ (v − αρ)(t0, p0, x0).

Thus v − ψ has a maximum at (t0, p0, x0) where

ψ(t, p, x) := αρ(t, p, x) + (v − αρ)(t0, p0, x0).

We now show that t0 = T . If not, since v is a viscosity subsolution to (21),we have

∂ψ

∂t(t0, p0, x0) +H+(D2ψ(t0, p0, x0), Dψ(t0, p0, x0), p0, x0) ≥ 0.

The left hand side of the preceding inequality is nothing but

α

(∂ρ

∂t(t0, p0, x0) +H+(D2ρ(t0, p0, x0), Dρ(t0, p0, x0), p0, x0)

),

14

which is strictly negative by Lemma 3.8. We thus have a contradiction.Therefore, t0 = T . By assumption we have v(T, p, x) ≤ 0, we thus have

v(t, p, x) ≤ αρ(t, p, x) in [t1, T )× Rn+1.

Letting α tend to zero, we get

v(t, p, x) ≤ 0 in [t1, T )× Rn+1.

We apply successively the same argument on the interval [t2, t1], where t2 :=(t1 − A/C1)

+ and if t2 > 0 we continue... We finally obtain

v(t, p, x) ≤ 0 in [0, T )× Rn+1.

That ends the proof.

3.2.3 The Dynamic Programming Principle

The Dynamic Programming Principle holds for V k defined as in Subsec-tion 3.2.1 (see Fleming & Souganidis [10, Thm.1.6]). Thus

V k(θ, p, x) = infΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x[Vk(t, P k(t), Xk(t))]. (27)

In order to get (10), we now let k tend to infinity. We first observe that, inview of (20), V k(θ, p, x) tends to V (θ, p, x). Second, we have V k(θ, p, x) =V k(θ, p, x) for all (p, x) ∈ Bk+1. Third, if (p, x) is in Bk+1 then (P k(t), Xk(t))is in Bk+1 for all t ∈ [θ, T ] almost surely. Thus, for all (p, x), one has

| infΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x[Vk(t, P k(t), Xk(t))] (28)

− infΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x[V (t, P (t), X(t))]|

≤ supΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x|V k(t, P k(t), Xk(t))− V (t, P (t), X(t))|

≤ supΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x|V k(t, P k(t), Xk(t))− V (t, P k(t), Xk(t))|

+ supΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x|V (t, P k(t), Xk(t))− V (t, P (t), X(t))|

≤ supΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x|(V k(t, P k(t), Xk(t))− V (t, P k(t), Xk(t)))Iτk≥T |

+ supΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x|(V k(t, P k(t), Xk(t))− V (t, P k(t), Xk(t)))Iτk<T |

+ supΠ∈AdΠ(θ)

supu(·)∈Adu(θ)

E θ,p,x|V (t, P k(t), Xk(t))− V (t, P (t), X(t))|.

(29)

15

In view of (20) and (18), we have

E θ,p,x|(V k(t, P k(t), Xk(t))− V (t, P k(t), Xk(t)))Iτk≥T |

≤ E θ,p,x

[K(|P k(t)|, |Xk(t)|, T )

√|P k(t)|2 + (Xk(t))2

k

]

≤ C(|p|, |x|, T )

k.

In addition, in view of (19),

E θ,p,x|V (t, P k(t), Xk(t))− V (t, P (t), X(t))|= E θ,p,x[|V (t, P k(t), Xk(t))− V (t, P (t), X(t))|Iτk<T ]

≤ K(|p|, |x|, T )

k

and

E θ,p,x|(V k(t, P k(t), Xk(t))− V (t, P k(t), Xk(t)))Iτk<T | ≤K(|p|, |x|, T )

k.

That ends the proof.

4 Example: worst case model risk manage-

ment for bond options

In this section we illustrate our result by considering the model risk controlwhen the trader aims to hedge a European option. The case of options onstocks readily appears as a particular case of the general setting introducedin the preceding sections. We thus describe a less elementary application,namely the case of options on bonds.

4.1 The Heath-Jarrow-Morton model

Suppose that the yield curve of the financial market follows the Heath-Jarrow-Morton (HJM) [11] model, and therefore we have:

Definition 4.1. For all time T ∗, the instantaneous forward rate f(t, T ∗)satisfies the stochastic differential equation

f(t, T ∗) = f(0, T ∗) +

∫ t

0

α(s, T ∗)ds+

∫ t

0

σ(s, T ∗)dWs, (30)

16

for a given Borel measurable function f(0, ·) : R+ → R, and given randommaps α : C × Ω → R, σ : C × Ω → R, where C := ((s, t) | 0 ≤ s ≤ t). Wesuppose that α(·, T ∗) and σ(·, T ∗) are adapted processes such that∫ T ∗

0

|α(s, T ∗)|ds+

∫ T ∗

0

|σ(s, T ∗)|2ds <∞ P − a.s.

When the market has no arbitrage, there exists an adapted process (λ(t))such that

α∗(t, T ∗) =1

2|σ(t, T ∗)|2 − σ∗(t, T ∗)λ(t),

where

α∗(t, T ∗) :=

∫ T ∗

t

α(t, s)ds

(see, e.g., Musiela & Rutkowski [13, Chap.13]).The HJM model is an infinite dimensional factor model. Special cases

are the one factor Merton model, the Vasicek model, the Cox-Ingersoll-Rossmodel, the Long & Schwartz model, the Hull & White model, the Ho & Leemodel, etc.

The price of a discount bond maturing at the date T ∗ is

B(t, T ∗) = exp

(−∫ T ∗

t

f(t, s)ds

). (31)

As shown in Musiela & Rutkowski [13, Lemma 13.1.1], the discount bondprice B(t, T ∗) satisfies

B(t, T ∗) = 1−∫ T ∗

t

(r(s) + σ∗(s, T ∗)λ(s))B(s, T ∗)ds

+

∫ T ∗

t

σ∗(s, T ∗)B(s, T ∗)dWs

for all 0 ≤ t ≤ T ∗, where r(t) := f(t, t) is the short term rate.In order to simplify notation we use forward prices. Given a price Pt,

define the forward price P Ft by

P Ft :=

Pt

B(t, TO),

where B(t, TO) is the price of the bond of maturity TO in the model drivenby σ(t, TO). In particular, the forward price of the discount bond maturingat T is

BF (t, T ) :=B(t, T )

B(t, TO).

17

It is easy to check that BF (t, T ) satisfies

dBF (t, T ) = BF (t, T )(σ∗(t, TO)− λ(t))(σ∗(t, TO)− σ∗(t, T ))dt

+BF (t, T )(σ∗(t, TO)− σ∗(t, T ))dWt.(32)

4.2 Exact hedging strategies

We consider the following situation. A trader wants to hedge a Europeanoption written on B(TO, T ). The payoff at maturity TO is denoted byf(B(TO, T )). The trader uses two bonds to hedge the option: the bondof maturity TO and the bond of maturity T .

At each date 0 ≤ t ≤ TO, a self-financing strategy consists in buying orselling a quantity HO

t of discount bonds of maturity TO and a quantity Ht

of discount bonds of maturity T such that

(i) The portfolio is self-financing, which means that, if

Vt = HtB(t, T ) +HOt B(t, TO) (33)

is the value of portfolio at time t, then

Vt = V0 +

∫ t

0

HsdB(s, T ) +

∫ t

0

HOs dB(s, TO). (34)

(ii) The processes (Ht) and (HOt ) are adapted and satisfy technical assump-

tions which ensure that all the stochastic integrals in the sequel, where(Ht) and (HO

t ) are integrated, are well defined and are martingales.

Let V Ft denote the forward price of the trader’s portfolio:

V Ft :=

Vt

B(t, TO).

For any self-financing strategy we have

dV Ft = HtdB

F (t, T ) (35)

(see Bossy et al. [5] for the proof). An exact hedging (replicating) strategyis a self-financing strategy (H,HO) such that

VT O = HT OB(TO, T ) +HOT O = f(B(TO, T )).

For a HJM model with a deterministic parameter σ, the exact hedging strat-egy satisfies

Ht =∂πσ

∂x(t, BF (t, T )),

18

where πσ is the solution to∂πσ

∂t(t, x) +

1

2x2(σ∗(t, TO)− σ∗(t, T ))2∂

2πσ

∂x2(t, x) = 0,

πσ(T, x) = f(x).

The quantity HOt is chosen according to the self-financing condition: in view

of (33) and (34) one has

HOt = V F

0 +

∫ t

0

HsdBF (s, T )−HtB

F (t, T ).

For details, see Bossy et al. [5].

4.3 Model risk

Bossy et al. [5] propose examples of model risk analysis based on Monte Carlosimulations of the Profit & Loss of self-financing misspecified strategies. Inthis subsection we give an outline of their results.

Suppose that the trader does not know the random map σ(s, T ). Instead,he or she chooses a deterministic model structure σ(s, T ) and tries to hedgethe contingent claim according to this model. The quantity H t of bonds ofmaturity T is determined according to the rule

H t =∂πσ

∂x(t, BF (t, T )),

where πσ is the solution to∂πσ

∂t(t, x) +

1

2x2(σ∗(t, TO)− σ∗(t, T ))2∂

2πσ

∂x2(t, x) = 0,

πσ(T, x) = f(x).

As above, the quantityHO

t is then determined by the self-financing condition.Let V t be the value of the trader’s portfolio at time t. The option seller’s

Profit & Loss at time t < TO is

P&Lt := V t − Vt,

and the forward price of the Profit & Loss is

P&LFt :=

P&Lt

B(t, TO).

19

One can study the probability law of the Profit & Loss at maturity TO, andcompute some of its statistics (moments, quantiles, etc.) by fixing σ(·, ·) andσ(·, ·). See Bossy et al. [5]. This methodology is however restricted to thecomparison of one (potentially incorrect) model against one or several (pos-sible true) models among a class of univariate Markov term structure models(with deterministic parameters). This class does not contain all possible termstructure models.

Consider the quantity

F(BF (TO, T ), V F

T O

):= E [G(f(BF (TO, T ))− V F

T O)],

where G is a given continuous function. It might be useful to choose thefunction G in order to be able to justify that our game problem correspondsto a kind of a measure of risk (to be defined). Of course such a measureof risk would be weaker than the coherent measures of risks introduced byArtzner, Delbaen, Eber and Heath [1]: for a discussion on a related topics,see Section 2 in [7] (where it must be understood that C/S0(T ) plays therole of f(BF (TO, T ))).

We assume that the function F (p, x) := G(f(p) − x) satisfies (8). Thecontrol processes of the market are

u1(t) := σ∗(t, TO)− λ(t),

u2(t) := σ∗(t, T )− σ∗(t, TO).

The control process of the trader is π(t) := HtBF (t,T )V F (t)

if V F (t) 6= 0 and π(t) :=

0 if V F (t) = 0. In view of (32) and (35), (P (t), X(t)) := (BF (t, T ), V F (t))satisfies a S.D.E of type (3) with r = 0, namely,

dP (t) = P (t)[u1(t)u2(t)dt+ u2(t)dWt],

dX(t) = X(t)π(t) [u1(t)u2(t)dt+ u2(t)dWt] .

The Hamilton-Jacobi-Bellman-Isaacs equation (9) then writes∂v

∂t(t, p, x) +H−(D2v(t, p, x), Dv(t, p, x), p, x) = 0 in [0, TO)× R2,

v(TO, p, x) = G(f(p)− x),

(36)where

H−(A, z, p, x) := maxu∈Ku

minπ∈Kπ[

1

2u2

2p2A11 + u2

2pxπA12 +1

2u2

2x2π2A22 + z1u1u2p+ z2u1u2πx

],

for all 2× 2 symmetric matrix A and all vector z in R2.

20

4.4 A numerical illustration

We finally illustrate that the value functions of our stochastic game problems,and the corresponding optimal strategies, can be numerically approximated.In our numerical example below, except for ρ = 0, the function F does notsatisfy the requirements for a coherent measure of risk (again see Remark2.3 in [7]); this is to emphasize the flexibility of our approach. We considera call option and set

F (p, x) := ((p−K)+ − x)+ + ρ(((p−K)+ − x)+)2.

The maturity TO of the option is 6 months and the option is written on adiscount bond of maturity T equal to 5 years. The trader uses two bonds tohedge this option: the bond of maturity 6 months and the bond of maturity 5years. The strike of the option is K = 0.509156. We set ρ = 20, π ∈ [−1, 1],u2 ∈ [0, 0.6] and u1 ∈ [−0.08, 0.06].

The finite difference method is used for the numerical resolution of the vis-cosity solution to (36) numerically. Because (36) is strongly degenerate, themonotone condition is not satisfied in the standard explicit finite differenceapproximation. We apply Kushner’s technique to overcome this difficulty,that is, there exists a (p, x, t, u1, u2, π)- dependent scale transformation wichyields a locally consistent approximation at (p, x, t, u1, u2, π). For details, seeKushner [14].

Our approximate value function is drawn in Fig.1. For other numericalresults (e.g., strategies obtained through the numerical resolution of the HJBIequation) and comments, see Talay & Zheng [16].

5 Conclusion

In this paper we have studied a stochastic game problem related to a worstcase model risk management. We have proven that the value function ofthis game problem is the unique viscosity solution to an Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, and satisfies the Dynamic ProgrammingPrinciple.

Empirical studies should be developed to check whether the strategies ob-tained by solving the HJBI equation numerically are realistic in practice. Thequestion of the accuracy of the numerical method should also be addressed.

21

Acknowledgement

This study has been done within the Risklab project ‘Model Risk’, which isa joint collaboration between the Risklab Institute in Zurich and the Omegaresearch group at Inria Sophia Antipolis. The authors thank Risklab for thefinancial support.

We also thank an anonymous referee for a suggestion which allowed us tosubstantially improve the presentation of the paper.

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Figure 1: The Value Function: Graph of V (0, p, x)

24